Question

Each month a store owner can spend at most 100,000$onP{C}^{\prime }slaptops.APCcoststhestoreowner$ $1000$ and a laptop costs him 1500$.EachPCissoldforaprofitof$ $400$ while laptop is sold for a profit of 700$.Thestoreownerestimatesthatatleast$15$P{C}^{\prime }sbutnomorethan$80$ are sold each month. He also estimates that the number of laptops sold is at most half the PC's. How many PC's and how many laptops should be sold in order to maximize the profit?

Answer 1

Let $x$ and $y$ be the numbers of PC's and laptops respectively that should be sold.
Profit $=400x+700y$ to maximize
Constraints
$15\le x\le 80$ least 15 PC's but no more than $80$ are sold each month
$y\le \left(1/2\right)x$
$1000x+1500y\le 100,000$ "store owner can spend at most 100,000$ on PC's and laptops"
$\{\begin{array}{c}15\le x\le 80\\ y\ge 0\\ y\le \left(1/2\right)x\\ 1000x+1500y\le 100,000\end{array}$
The solution set of the system of inequalities above and the vertices of the feasible solution set obtained are shown.
Vertices:
$A$ at intersection of $x=15$ and $y=0$ (x-axis) coordinates of $A:(15,0)$
$B$ at intersection of $x=15$ and $y=\left(1/2\right)x$ coordinates of $B:(15,7.5)$
$C$ at intersection of $y=\left(1/2\right)x$ and $1000x+1500y=100000$ coordinates of $C:(57.14,28.57)$
$D$ at intersection of $1000x+1500y=100000$ and $x=80$ (y-axis) coordinates of $D:(80,13,3)$
Evaluate the profit at each vertex
$A(15,0),P=400\times 15+700\times 0=6000$
$B(15,7.5).P=400\times 15+700\times 7.5=11250$
$C(57.14,28.57),P(400\times 57.14+700\times 28.57=42855$
$D(80,13.3),P=400\times 80+700\times 13.3=41310$
The profit is maximum for $x=57.14$ and $y=28.57$ but these cannot be accepted as solutions because $x$ and $y$ are numbers of PC's and ;laptops and must be integers. We need to select the nearest integers to $x=57.14$ and $y=28.57$ that are satisfy all constraints and give a maximum profit
$x=57$ and $y=29$ do not satisfy all constraints
$x=57$ and $y=28$ satisfy all constraints
Profit$=500\times 57+700\times 28=42400$, which is maximum